Linear, Tensor, and Graded Algebra

The algebra of fields begins with a simple question: which parts of a component calculation describe the object itself? This chapter develops finite-dimensional vectors, tensors, adjoints, spectral projectors, and graded signs so that changes of basis do not become changes of physics.

Enter this chapter

Begin with Vector Spaces, Duals, Linear Maps, and Bases to distinguish an invariant map from its matrix. For fermionic calculations, continue through tensor products to Grassmann variables.

Topics

1. Vector Spaces, Duals, Linear Maps, and Bases — Separates vectors, covectors, and maps from their basis-dependent components.
2. Direct Sums, Tensor Products, and Index Structure — Distinguishes independent sectors from multilinear composites and their contractions.
3. Bilinear and Hermitian Forms, Adjoints, and Isometries — Explains how a chosen form determines orthogonality, adjoints, and norm preservation.
4. Normal Forms, Spectra, and Projectors — Distinguishes diagonalization, orthogonal spectral resolution, Jordan form, and singular values.
5. Commutators and Operator Exponentials — Uses nested commutators to control conjugation, exponential products, and ordering.
6. Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration — Tracks parity signs and derives determinant and Pfaffian Gaussian integrals.

How the ideas fit together

A field multiplet and its source first require a vector space and its dual; their pairing needs no metric. A Hermitian form adds an adjoint, and normality then permits orthogonal spectral projectors. Thus a basis choice, an inner product, and a decomposition into modes are different pieces of information. Diagonalizability alone does not guarantee an orthonormal eigenbasis; see Axler 2024, §§ 3F, 6A, and 7A.

Grading adds another independent structure: exchanging two odd factors changes a sign. It does not turn finite matrices into canonical position and momentum operators. Infinite-dimensional representations require additional attention to domains.

Review the chapter

Why can finite matrices not satisfy $[Q,P]=iI$ on a nonzero complex vector space?

Solution

Cyclicity gives $\operatorname{tr}(QP-PQ)=0$, while $\operatorname{tr}(iI)=i\dim V\ne0$. The contradiction identifies a finite-dimensional obstruction; it does not establish that arbitrary unbounded operators satisfy the relation.

Where to go next

Continue to Functional and Spectral Analysis when domains and continuous spectra enter, or Lie Groups, Representations, and Spinors for symmetry transformations.

References

• Sheldon Axler, Linear Algebra Done Right, 4th ed., Graduate Texts in Mathematics 250, Springer, 2024. DOI.